derivative of ((x-2(x^2-x+1))/(sqrt(x)))

解

\frac{d}{d x} (\frac{( x - 2 ) ( x ^{2} - x + 1 )}{x})

\frac{5 x ^{3} - 9 x ^{2} + 3 x + 2}{2 x x}

解答ステップ

\frac{d}{d x} (\frac{( x - 2 ) ( x ^{2} - x + 1 )}{x})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ( x - 2 ) ( x ^{2} - x + 1 ) ) x - \frac{d}{d x} ( x ) ( x - 2 ) ( x ^{2} - x + 1 )}{( x ) ^{2}}

\frac{d}{d x} ((x - 2) (x^{2} - x + 1)) = 3 x^{2} - 6 x + 3

\frac{d}{d x} (x) = \frac{1}{2 x}

= \frac{( 3 x ^{2} - 6 x + 3 ) x - \frac{1}{2 x} ( x - 2 ) ( x ^{2} - x + 1 )}{( x ) ^{2}}

簡素化

\frac{( 3 x ^{2} - 6 x + 3 ) x - \frac{1}{2 x} ( x - 2 ) ( x ^{2} - x + 1 )}{( x ) ^{2}} : \frac{5 x ^{3} - 9 x ^{2} + 3 x + 2}{2 x x}

= \frac{5 x ^{3} - 9 x ^{2} + 3 x + 2}{2 x x}